Earth Swallowing black hole?

The Nomad · Post by **The Nomad** » 2006-11-15 04:44pm

I'm looking for an estimation of the minimum mass of a black hole that could, if dropped at the center of the Earth, actually swallow it in around a decade.

A black hole with a ten years lifespan would have a mass of around 1 billion tons, a radius of around 1.5 femtometers, and a Hawking luminosity of 350 MW.

Now could such a black hole actually swallow enough material to maintain its mass and sustain its growth (against it own tendency to evaporate through Hawking radiation) and eventually swallow the whole planet? The fucker emits a few hundred megawatts, but I doubt this will be enough to counteract the pull of its gravity. But won't the gravity tides around it overheat matter to the point that said matter expands and escapes its gravity?

The Nomad · Post by **The Nomad** » 2006-11-15 04:49pm

Holy shit, I totally fucked up the numbers

It should read : about 150,000 tons, 200 zeptometers, and around 15 petawatts.

I guess such power output is much more likely to push any matter capable of feeding it away, though.

Frank Hipper · Post by **Frank Hipper** » 2006-11-15 06:13pm

A black hole with a mass of 150,000 tons?

Are you serious?

Sriad · Post by **Sriad** » 2006-11-15 06:24pm

By my own (fairly off the cuff) calculations, a 150,000 ton BH would only last around 40 days. (Sriad's Black Hole Lifetime Equation for Ameteurs is: [mass (in tons)/1000]^3=lifetime in seconds)

A ten year hole would mass 680,000 tons, still extremely hot and small.

Gravity tides are actually how black holes (even real ones) get matter to actually fall into them instead of endlessly orbiting. When there's enough to form an accretion disk (or sphere, in the case of a BH in such a matter rich environment as the Earth's core) matter which is incrementally closer to the black hole must move incrementally faster than its outer neighbor. The difference in speed creates friction, thus heat, robbing the particles of KE and forcing them to fall into lower orbits until they're within 1.5 Schwarzschild radii and promptly gulped up. Any black hole falling through the Earth would be surrounded by an absurdly hot shell of infalling matter traveling with it, disrupting and being disrupted by the new stuff it encounters on its orbit through the core... The heat this generates (matter falling from flat space-time into a black hole radiates ~41% of its rest mass in heat during the falling-in process) would probably be more devastating to Earth than the missing matter.

Someone more sciency than I will probably be able to give you better numbers.

The Nomad · Post by **The Nomad** » 2006-11-16 01:18am

Ok I posted this just before going to sleep so I likely totally messed the calcs up

Wyrm · Post by **Wyrm** » 2006-11-16 01:45am

Hmmm, this is an interesting problem. See, not only does the black hole have to eat enough matter per second to balance out the mass radiated away through Hawking radiation, it must also be big enough so that Hawking radiation it produces is not intense enough to push infalling matter away from the black hole.

Winston Blake · Post by **Winston Blake** » 2006-11-17 08:33am

Sriad wrote:By my own (fairly off the cuff) calculations, a 150,000 ton BH would only last around 40 days. (Sriad's Black Hole Lifetime Equation for Ameteurs is: [mass (in tons)/1000]^3=lifetime in seconds)

A ten year hole would mass 680,000 tons, still extremely hot and small.

What are your equations? I get the same as Nomad. Anyway, I don't know if there's any simple way to find the answer to this.

Everybody, keep in mind that if the accretion rate passes the Eddington limit, the accretion disc models for big black holes become invalid. Assuming I've done this right, to eat the Earth in 10 years, a black hole not passing the Eddington limit would have to be 2.6e32 kg. Presumably a black hole inside the Earth would eat much faster. So all I can say is between 150e6 kg and 2.6e32 kg.

GrandMasterTerwynn · Post by **GrandMasterTerwynn** » 2006-11-17 10:04am

Winston Blake wrote:
Sriad wrote:By my own (fairly off the cuff) calculations, a 150,000 ton BH would only last around 40 days. (Sriad's Black Hole Lifetime Equation for Ameteurs is: [mass (in tons)/1000]^3=lifetime in seconds)

A ten year hole would mass 680,000 tons, still extremely hot and small.
What are your equations? I get the same as Nomad. Anyway, I don't know if there's any simple way to find the answer to this.

Everybody, keep in mind that if the accretion rate passes the Eddington limit, the accretion disc models for big black holes become invalid. Assuming I've done this right, to eat the Earth in 10 years, a black hole not passing the Eddington limit would have to be 2.6e32 kg. Presumably a black hole inside the Earth would eat much faster. So all I can say is between 150e6 kg and 2.6e32 kg.

Slight problem with the upper-limit number there. The Earth itself only masses 5.97E+24 kilograms. Your proposed black hole is about 100x more massive than the Sun, which has a mass of 1.99E+30 kilograms.

Winston Blake · Post by **Winston Blake** » 2006-11-17 02:10pm

GrandMasterTerwynn wrote:Slight problem with the upper-limit number there. The Earth itself only masses 5.97E+24 kilograms. Your proposed black hole is about 100x more massive than the Sun, which has a mass of 1.99E+30 kilograms.

So? The question is how big a black hole has to be for there to be no Earth left after 10 years. If it's about everybody dying from the gravity, eating only a tenth of the Earth's volume would still devastate the world, but that's not the question.

If we assume it only changes gravity to 1/4 extra to limit the global catastrophe, then the limit is 7.5e24 kg. It could go less if we have to consider its passage down into Earth.